Abundant Number
An abundant number, sometimes also called an excessive number, is a positive integer 
for which
 |
(1)
|
where 
is the divisor
function and 
is the restricted divisor function.
The quantity 
is sometimes called the abundance.
A number which is abundant but for which all its proper divisors are deficient is called a primitive
abundant number (Guy 1994, p. 46).
The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (OEIS A005101).
Every positive integer 
with 
is abundant.
Any multiple of a perfect number or an abundant
number is also abundant. Prime numbers are not abundant. Every number greater than
20161 can be expressed as a sum of two abundant numbers.
There are only 21 abundant numbers less than 100, and they are all even.
The first odd abundant number is
 |
(2)
|
That 945 is abundant can be seen by computing
 |
(3)
|
Define the density function
 |
(4)
|
(correcting the expression in Finch 2003, p. 126) for a positive real number 
where 
gives the cardinal number of the set 
, then Davenport
(1933) proved that 
exists and is continuous for all
, and Erdős (1934) gave a simplified proof
(Finch 2003). The special case 
then gives
the asymptotic density of abundant numbers,
 |
(5)
|
The following table summarizes improvements in bounds on the constant over time.
| value | reference |
 | Behrend (1933) |
 | Wall (1971) and Wall et al. (1977) |
 | Deléglise
(1998) |
 | Kobayashi (2010, p. 12) |
SEE ALSO: Abundance,
Aliquot Sequence,
Colossally Abundant Number,
Deficient Number,
Highly
Composite Number,
Multiamicable Numbers,
Perfect Number,
Practical
Number,
Primitive Abundant Number,
Weird Number
REFERENCES:
Behrend, F. "Über numeri abundantes." Sitzungsber. Preuss. Akad.
Wiss., Phys.-Math. Kl., No. 21/23, 322-328, 1932.
Behrend, F. "Über numeri abundantes. II." Sitzungsber. Preuss.
Akad. Wiss., Phys.-Math. Kl., No. 6, 280-293, 1933.
Davenport, H. "Über numeri abundantes." Sitzungsber. Preuss. Akad.
Wiss., Phys.-Math. Kl., No. 6, 830-837, 1933.
Deléglise, M. "Bounds for the Density of Abundant Integers." Exp.
Math. 7, 137-143, 1998.
Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York:
Dover, pp. 3-33, 2005.
Erdős, P. "On the Density of the Abundant Numbers." J. London Math.
Soc. 9, 278-282, 1934.
Finch, S. R. "Abundant Numbers Density Constant." §2.11 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 126-127,
2003.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-46,
1994.
Kobayashi, M. "On the Density of Abundant Numbers." Ph.D. thesis. Hanover, NH: Dartmouth College, 2010.
Singh, S. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem.
New York: Walker, pp. 11 and 13, 1997.
Sloane, N. J. A. Sequence A005101/M4825
in "The On-Line Encyclopedia of Integer Sequences."
Souissi, M. Un Texte Manuscrit d'Ibn Al-Bannā' Al-Marrakusi sur les Nombres Parfaits, Abondants, Deficients, et Amiables. Karachi, Pakistan: Hamdard Nat.
Found., 1975.
Wall, C. R. "Density Bounds for the Sum of Divisors Function." In The
Theory of Arithmetic Functions: Proceedings of the Conference at Western Michigan
University, April 29-May 1, 1971. (Ed. A. A. Gioia and D. L. Goldsmith).
New York: Springer-Verlag, pp. 283-287, 1971.
Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density Bounds for the Sum of Divisors Function." Math. Comput. 26, 773-777,
1972.
Wall, C. R.; Crews, P. L.; and Johnson, D. B. "Density Bounds
for the Sum of Divisors Function." Math. Comput. 31, 616, 1977.
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Abundant Number
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Weisstein, Eric W. "Abundant Number."
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